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%I #20 Apr 03 2021 19:04:31
%S 49,81,169,361,961,1369,1849,3721,3969,4489,5329,6241,8281,9409,10609,
%T 11881,13689,16129,17689,19321,22801,24649,26569,29241,32761,37249,
%U 39601,44521,47089,49729,52441,58081,61009,67081,73441,76729,77841,80089,90601,94249,97969
%N Discriminants of cyclic cubic fields.
%C Square terms in A006832.
%C Numbers of the form k^2 where A160498(k) >= 2.
%C Each term k^2 is associated with A343003(k) cyclic cubic fields.
%C Let D be a discriminant of a cubic field F, then F is a cyclic cubic field if and only if D is a square. For D = k^2, k must be of the form (p_1)*(p_2)*...*(p_t) or 9*(p_1)*(p_2)*...*(p_{t-1}) with distinct primes p_i == 1 (mod 3), in which case there are exactly 2^(t-1) = 2^(omega(k)-1) (cyclic) cubic fields with discriminant D. See Page 17, Theorem 2.7 of the Ka Lun Wong link.
%H Jianing Song, <a href="/A343000/b343000.txt">Table of n, a(n) for n = 1..3200</a>
%H LMFDB, <a href="https://www.lmfdb.org/NumberField/?degree=3">Cubic fields</a>
%H Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Cubic_field">Cubic field</a>
%H Ka Lun Wong, <a href="https://scholarsarchive.byu.edu/etd/2781">Maximal Unramified Extensions of Cyclic Cubic Fields</a>, (2011), Theses and Dissertations, 2781.
%F a(n) = A343001(n)^2.
%e 49 = 7^2 is a term since it is the discriminant of the cyclic cubic field Q[x]/(x^3 - x^2 + x + 1).
%e 81 = 9^2 is a term since it is the discriminant of the cyclic cubic field Q[x]/(x^3 - 3x - 1).
%o (PARI) isA343000(n) = if(issquare(n)&&n>1, my(k=sqrtint(n), L=factor(k), w=omega(k)); for(i=1, w, if(!((L[i, 1]%3==1 && L[i, 2]==1) || L[i, 1]^L[i, 2] == 9), return(0))); 1)
%Y Discriminants and their square roots of cyclic cubic fields:
%Y At least 1 associated cyclic cubic field: this sequence, A343001.
%Y Exactly 1 associated cyclic cubic field: A343022, A002476 U {9}.
%Y At least 2 associated cyclic cubic fields: A343024, A343025.
%Y Exactly 2 associated cyclic cubic fields: A343002, A343003.
%Y Cf. A006832, A160498, A343023.
%K nonn,easy
%O 1,1
%A _Jianing Song_, Apr 02 2021
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